solid mode vs. sculpt mode
To answer the first part of your question, any mesh can be sculpted on in Blender. The way that sculpting works is by moving the positions of vertices, so your model must have vertices where you are sculpting otherwise nothing will happen. 

For example, you will not be able to sculpt any detail onto the default cube, because it consists of only 8 total vertices. 


But, if a Subdivision Surface or Multiresolution modifier is added, the model will gain sufficient geometry to be easily sculptable.



As for having multiple vertices in the same location, this can be resolved by pressing W > Remove Doubles in edit mode (in 2.79).